
where \((a)\)  stems from the Be-the-Leader (BTL) Lemma [1]} applied with \(x^\star \!\in \! \mathcal {X}_T\!\subseteq \! \mathcal {X}\)  to (REF ); and \((b)\)  from expanding \(\mathcal {L}_t(x^\star , \lambda _t)\) , using \(g_t(x^\star ) \!\preceq 0\)  and \(\Vert x^\star \!-x_t\Vert ^2\!\le \! 4D^2, \forall t\) . Add \(\sum _{t=1}^{T}c_t^\top x_t\)  to both sides and rearrange:
\(\!\!\!\!\sum _{t=1}^{T}\! c_t^\top (x_t\!-\!x^\star )\!\le \! 2D^2\sigma _{1:T}\!+\!\sum _{t=1}^{T}\!\frac{\xi _t^2}{\phi _{0:t-1}}\!+ c_t^\top \big (x_t\!-\!z_t\big )\!\!\) 
